I know that if $K$ is a field then $K[x]$ is a Euclidean domain and every Euclidean domain is a PID. In this way I can prove that $K[x]$ is a PID. But is there a method to show $K[x]$ is a PID directly from the definition? I mean a usual procedure is to design […]

Let $p$ be a prime number. Then if $ f(x) = (1+x)^p$ and $g(x) = (1+x)$, then is $f \equiv g \mod p$? I’m trying to prove that for integers $a > b > 0$ and a prime integer $p$, ${pa\choose b} \equiv {a \choose b}.$ To do this I use FLT to show that […]

Prove that an Infinite group must have subgroup with infinite elements. I know that if group was cyclic order of the generator is infinite and there are infinite number of divisors.

I know that if $f$ is a polynomial over a subfield $F$ of $\mathbb{C}$ and $f$ is solvable by radicals, then the Galois group of $f$ over $F$ is solvable. I’ve also seen many applications of this fact in demonstrating that certain quintic polynomials over $\mathbb{Q}$ are not solvable by radicals. My question is this: […]

We know the fact that: If a module $M$ is finitely cogenerated, then every module that cogenerates $M$ finitely cogenerates $M$. Conversely, it is not true. I find an example in the book “Rings and Categories of modules” written by Frank W. Anderson and Kent R. Fuller. Example The abelian group $\bigoplus_{\mathbb{P}}{\mathbb{Z}_p}$, ($p$ is a […]

The algebraic equations in one variable, in the general case, cannot be solved by radicals. While the basic operations and root extraction applied to the coefficients of the equations of degree $2, 3, 4 $ are sufficient to express the general solution, for arbitrary degree equations, the general solution can be expressed only in terms of much […]

I have some true or false questions and would like to have your help to check on it. A). in a ring R, if $x^2=x$, $\forall x\in R$, then R is commutative For (A), when looking at $(x+y)^2$, it has $x+y=(x+y)^2=x^2+xy+yx+y^2$ and then yx+xy=0, and from 2x=4x, therefore 2x=0. how this play a role here? […]

I am trying to study some module theory using the book “Algebra” by Michael Artin (2nd Edition, to be precise), and I can’t really fathom what is written in Section 14.5. Left multiplication by an $m \times n$ matrix defines a homomorphism of $R$-modules $A: R^n \rightarrow R^m$. Its image consists of all linear combinations […]

There is in module theory the following (krull-shmidt) theorem: If $(M_i)_{i\in I}$ and $(N_j)_{j\in J}$ are two families of simple module such that $\bigoplus\limits_{i\in I} M_i \simeq \bigoplus\limits_{j\in J} M_j$, then there exists a bijection $\sigma:I\rightarrow J$ such that $M_i \simeq N_{\sigma(i)}$. Is there a similar kind of theorem for partially ordered sets? More precisely, […]

How can we prove that the vector space of polynomials in one variable, $\mathbb{R}[x]$ is not finite dimensional?

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